3.333 \(\int \frac {1}{\sqrt {x} (1+x^2)^3} \, dx\)

Optimal. Leaf size=129 \[ \frac {7 \sqrt {x}}{16 \left (x^2+1\right )}+\frac {\sqrt {x}}{4 \left (x^2+1\right )^2}-\frac {21 \log \left (x-\sqrt {2} \sqrt {x}+1\right )}{64 \sqrt {2}}+\frac {21 \log \left (x+\sqrt {2} \sqrt {x}+1\right )}{64 \sqrt {2}}-\frac {21 \tan ^{-1}\left (1-\sqrt {2} \sqrt {x}\right )}{32 \sqrt {2}}+\frac {21 \tan ^{-1}\left (\sqrt {2} \sqrt {x}+1\right )}{32 \sqrt {2}} \]

[Out]

21/64*arctan(-1+2^(1/2)*x^(1/2))*2^(1/2)+21/64*arctan(1+2^(1/2)*x^(1/2))*2^(1/2)-21/128*ln(1+x-2^(1/2)*x^(1/2)
)*2^(1/2)+21/128*ln(1+x+2^(1/2)*x^(1/2))*2^(1/2)+1/4*x^(1/2)/(x^2+1)^2+7/16*x^(1/2)/(x^2+1)

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Rubi [A]  time = 0.07, antiderivative size = 129, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 8, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.615, Rules used = {290, 329, 211, 1165, 628, 1162, 617, 204} \[ \frac {7 \sqrt {x}}{16 \left (x^2+1\right )}+\frac {\sqrt {x}}{4 \left (x^2+1\right )^2}-\frac {21 \log \left (x-\sqrt {2} \sqrt {x}+1\right )}{64 \sqrt {2}}+\frac {21 \log \left (x+\sqrt {2} \sqrt {x}+1\right )}{64 \sqrt {2}}-\frac {21 \tan ^{-1}\left (1-\sqrt {2} \sqrt {x}\right )}{32 \sqrt {2}}+\frac {21 \tan ^{-1}\left (\sqrt {2} \sqrt {x}+1\right )}{32 \sqrt {2}} \]

Antiderivative was successfully verified.

[In]

Int[1/(Sqrt[x]*(1 + x^2)^3),x]

[Out]

Sqrt[x]/(4*(1 + x^2)^2) + (7*Sqrt[x])/(16*(1 + x^2)) - (21*ArcTan[1 - Sqrt[2]*Sqrt[x]])/(32*Sqrt[2]) + (21*Arc
Tan[1 + Sqrt[2]*Sqrt[x]])/(32*Sqrt[2]) - (21*Log[1 - Sqrt[2]*Sqrt[x] + x])/(64*Sqrt[2]) + (21*Log[1 + Sqrt[2]*
Sqrt[x] + x])/(64*Sqrt[2])

Rule 204

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> -Simp[ArcTan[(Rt[-b, 2]*x)/Rt[-a, 2]]/(Rt[-a, 2]*Rt[-b, 2]), x] /
; FreeQ[{a, b}, x] && PosQ[a/b] && (LtQ[a, 0] || LtQ[b, 0])

Rule 211

Int[((a_) + (b_.)*(x_)^4)^(-1), x_Symbol] :> With[{r = Numerator[Rt[a/b, 2]], s = Denominator[Rt[a/b, 2]]}, Di
st[1/(2*r), Int[(r - s*x^2)/(a + b*x^4), x], x] + Dist[1/(2*r), Int[(r + s*x^2)/(a + b*x^4), x], x]] /; FreeQ[
{a, b}, x] && (GtQ[a/b, 0] || (PosQ[a/b] && AtomQ[SplitProduct[SumBaseQ, a]] && AtomQ[SplitProduct[SumBaseQ, b
]]))

Rule 290

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> -Simp[((c*x)^(m + 1)*(a + b*x^n)^(p + 1))/(
a*c*n*(p + 1)), x] + Dist[(m + n*(p + 1) + 1)/(a*n*(p + 1)), Int[(c*x)^m*(a + b*x^n)^(p + 1), x], x] /; FreeQ[
{a, b, c, m}, x] && IGtQ[n, 0] && LtQ[p, -1] && IntBinomialQ[a, b, c, n, m, p, x]

Rule 329

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> With[{k = Denominator[m]}, Dist[k/c, Subst[I
nt[x^(k*(m + 1) - 1)*(a + (b*x^(k*n))/c^n)^p, x], x, (c*x)^(1/k)], x]] /; FreeQ[{a, b, c, p}, x] && IGtQ[n, 0]
&& FractionQ[m] && IntBinomialQ[a, b, c, n, m, p, x]

Rule 617

Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*Simplify[(a*c)/b^2]}, Dist[-2/b, Sub
st[Int[1/(q - x^2), x], x, 1 + (2*c*x)/b], x] /; RationalQ[q] && (EqQ[q^2, 1] ||  !RationalQ[b^2 - 4*a*c])] /;
FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 628

Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[(d*Log[RemoveContent[a + b*x +
c*x^2, x]])/b, x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]

Rule 1162

Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[(2*d)/e, 2]}, Dist[e/(2*c), Int[1/S
imp[d/e + q*x + x^2, x], x], x] + Dist[e/(2*c), Int[1/Simp[d/e - q*x + x^2, x], x], x]] /; FreeQ[{a, c, d, e},
x] && EqQ[c*d^2 - a*e^2, 0] && PosQ[d*e]

Rule 1165

Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[(-2*d)/e, 2]}, Dist[e/(2*c*q), Int[
(q - 2*x)/Simp[d/e + q*x - x^2, x], x], x] + Dist[e/(2*c*q), Int[(q + 2*x)/Simp[d/e - q*x - x^2, x], x], x]] /
; FreeQ[{a, c, d, e}, x] && EqQ[c*d^2 - a*e^2, 0] && NegQ[d*e]

Rubi steps

\begin {align*} \int \frac {1}{\sqrt {x} \left (1+x^2\right )^3} \, dx &=\frac {\sqrt {x}}{4 \left (1+x^2\right )^2}+\frac {7}{8} \int \frac {1}{\sqrt {x} \left (1+x^2\right )^2} \, dx\\ &=\frac {\sqrt {x}}{4 \left (1+x^2\right )^2}+\frac {7 \sqrt {x}}{16 \left (1+x^2\right )}+\frac {21}{32} \int \frac {1}{\sqrt {x} \left (1+x^2\right )} \, dx\\ &=\frac {\sqrt {x}}{4 \left (1+x^2\right )^2}+\frac {7 \sqrt {x}}{16 \left (1+x^2\right )}+\frac {21}{16} \operatorname {Subst}\left (\int \frac {1}{1+x^4} \, dx,x,\sqrt {x}\right )\\ &=\frac {\sqrt {x}}{4 \left (1+x^2\right )^2}+\frac {7 \sqrt {x}}{16 \left (1+x^2\right )}+\frac {21}{32} \operatorname {Subst}\left (\int \frac {1-x^2}{1+x^4} \, dx,x,\sqrt {x}\right )+\frac {21}{32} \operatorname {Subst}\left (\int \frac {1+x^2}{1+x^4} \, dx,x,\sqrt {x}\right )\\ &=\frac {\sqrt {x}}{4 \left (1+x^2\right )^2}+\frac {7 \sqrt {x}}{16 \left (1+x^2\right )}+\frac {21}{64} \operatorname {Subst}\left (\int \frac {1}{1-\sqrt {2} x+x^2} \, dx,x,\sqrt {x}\right )+\frac {21}{64} \operatorname {Subst}\left (\int \frac {1}{1+\sqrt {2} x+x^2} \, dx,x,\sqrt {x}\right )-\frac {21 \operatorname {Subst}\left (\int \frac {\sqrt {2}+2 x}{-1-\sqrt {2} x-x^2} \, dx,x,\sqrt {x}\right )}{64 \sqrt {2}}-\frac {21 \operatorname {Subst}\left (\int \frac {\sqrt {2}-2 x}{-1+\sqrt {2} x-x^2} \, dx,x,\sqrt {x}\right )}{64 \sqrt {2}}\\ &=\frac {\sqrt {x}}{4 \left (1+x^2\right )^2}+\frac {7 \sqrt {x}}{16 \left (1+x^2\right )}-\frac {21 \log \left (1-\sqrt {2} \sqrt {x}+x\right )}{64 \sqrt {2}}+\frac {21 \log \left (1+\sqrt {2} \sqrt {x}+x\right )}{64 \sqrt {2}}+\frac {21 \operatorname {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\sqrt {2} \sqrt {x}\right )}{32 \sqrt {2}}-\frac {21 \operatorname {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\sqrt {2} \sqrt {x}\right )}{32 \sqrt {2}}\\ &=\frac {\sqrt {x}}{4 \left (1+x^2\right )^2}+\frac {7 \sqrt {x}}{16 \left (1+x^2\right )}-\frac {21 \tan ^{-1}\left (1-\sqrt {2} \sqrt {x}\right )}{32 \sqrt {2}}+\frac {21 \tan ^{-1}\left (1+\sqrt {2} \sqrt {x}\right )}{32 \sqrt {2}}-\frac {21 \log \left (1-\sqrt {2} \sqrt {x}+x\right )}{64 \sqrt {2}}+\frac {21 \log \left (1+\sqrt {2} \sqrt {x}+x\right )}{64 \sqrt {2}}\\ \end {align*}

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Mathematica [A]  time = 0.03, size = 121, normalized size = 0.94 \[ \frac {1}{128} \left (\frac {56 \sqrt {x}}{x^2+1}+\frac {32 \sqrt {x}}{\left (x^2+1\right )^2}-21 \sqrt {2} \log \left (x-\sqrt {2} \sqrt {x}+1\right )+21 \sqrt {2} \log \left (x+\sqrt {2} \sqrt {x}+1\right )-42 \sqrt {2} \tan ^{-1}\left (1-\sqrt {2} \sqrt {x}\right )+42 \sqrt {2} \tan ^{-1}\left (\sqrt {2} \sqrt {x}+1\right )\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[1/(Sqrt[x]*(1 + x^2)^3),x]

[Out]

((32*Sqrt[x])/(1 + x^2)^2 + (56*Sqrt[x])/(1 + x^2) - 42*Sqrt[2]*ArcTan[1 - Sqrt[2]*Sqrt[x]] + 42*Sqrt[2]*ArcTa
n[1 + Sqrt[2]*Sqrt[x]] - 21*Sqrt[2]*Log[1 - Sqrt[2]*Sqrt[x] + x] + 21*Sqrt[2]*Log[1 + Sqrt[2]*Sqrt[x] + x])/12
8

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fricas [A]  time = 0.79, size = 173, normalized size = 1.34 \[ -\frac {84 \, \sqrt {2} {\left (x^{4} + 2 \, x^{2} + 1\right )} \arctan \left (\sqrt {2} \sqrt {\sqrt {2} \sqrt {x} + x + 1} - \sqrt {2} \sqrt {x} - 1\right ) + 84 \, \sqrt {2} {\left (x^{4} + 2 \, x^{2} + 1\right )} \arctan \left (\frac {1}{2} \, \sqrt {2} \sqrt {-4 \, \sqrt {2} \sqrt {x} + 4 \, x + 4} - \sqrt {2} \sqrt {x} + 1\right ) - 21 \, \sqrt {2} {\left (x^{4} + 2 \, x^{2} + 1\right )} \log \left (4 \, \sqrt {2} \sqrt {x} + 4 \, x + 4\right ) + 21 \, \sqrt {2} {\left (x^{4} + 2 \, x^{2} + 1\right )} \log \left (-4 \, \sqrt {2} \sqrt {x} + 4 \, x + 4\right ) - 8 \, {\left (7 \, x^{2} + 11\right )} \sqrt {x}}{128 \, {\left (x^{4} + 2 \, x^{2} + 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(x^2+1)^3/x^(1/2),x, algorithm="fricas")

[Out]

-1/128*(84*sqrt(2)*(x^4 + 2*x^2 + 1)*arctan(sqrt(2)*sqrt(sqrt(2)*sqrt(x) + x + 1) - sqrt(2)*sqrt(x) - 1) + 84*
sqrt(2)*(x^4 + 2*x^2 + 1)*arctan(1/2*sqrt(2)*sqrt(-4*sqrt(2)*sqrt(x) + 4*x + 4) - sqrt(2)*sqrt(x) + 1) - 21*sq
rt(2)*(x^4 + 2*x^2 + 1)*log(4*sqrt(2)*sqrt(x) + 4*x + 4) + 21*sqrt(2)*(x^4 + 2*x^2 + 1)*log(-4*sqrt(2)*sqrt(x)
+ 4*x + 4) - 8*(7*x^2 + 11)*sqrt(x))/(x^4 + 2*x^2 + 1)

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giac [A]  time = 0.61, size = 94, normalized size = 0.73 \[ \frac {21}{64} \, \sqrt {2} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} + 2 \, \sqrt {x}\right )}\right ) + \frac {21}{64} \, \sqrt {2} \arctan \left (-\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} - 2 \, \sqrt {x}\right )}\right ) + \frac {21}{128} \, \sqrt {2} \log \left (\sqrt {2} \sqrt {x} + x + 1\right ) - \frac {21}{128} \, \sqrt {2} \log \left (-\sqrt {2} \sqrt {x} + x + 1\right ) + \frac {7 \, x^{\frac {5}{2}} + 11 \, \sqrt {x}}{16 \, {\left (x^{2} + 1\right )}^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(x^2+1)^3/x^(1/2),x, algorithm="giac")

[Out]

21/64*sqrt(2)*arctan(1/2*sqrt(2)*(sqrt(2) + 2*sqrt(x))) + 21/64*sqrt(2)*arctan(-1/2*sqrt(2)*(sqrt(2) - 2*sqrt(
x))) + 21/128*sqrt(2)*log(sqrt(2)*sqrt(x) + x + 1) - 21/128*sqrt(2)*log(-sqrt(2)*sqrt(x) + x + 1) + 1/16*(7*x^
(5/2) + 11*sqrt(x))/(x^2 + 1)^2

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maple [A]  time = 0.01, size = 86, normalized size = 0.67 \[ \frac {21 \sqrt {2}\, \arctan \left (\sqrt {2}\, \sqrt {x}-1\right )}{64}+\frac {21 \sqrt {2}\, \arctan \left (\sqrt {2}\, \sqrt {x}+1\right )}{64}+\frac {21 \sqrt {2}\, \ln \left (\frac {x +\sqrt {2}\, \sqrt {x}+1}{x -\sqrt {2}\, \sqrt {x}+1}\right )}{128}+\frac {\sqrt {x}}{4 \left (x^{2}+1\right )^{2}}+\frac {7 \sqrt {x}}{16 \left (x^{2}+1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x^2+1)^3/x^(1/2),x)

[Out]

1/4*x^(1/2)/(x^2+1)^2+7/16/(x^2+1)*x^(1/2)+21/128*2^(1/2)*ln((x+2^(1/2)*x^(1/2)+1)/(x-2^(1/2)*x^(1/2)+1))+21/6
4*2^(1/2)*arctan(2^(1/2)*x^(1/2)+1)+21/64*2^(1/2)*arctan(2^(1/2)*x^(1/2)-1)

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maxima [A]  time = 2.90, size = 99, normalized size = 0.77 \[ \frac {21}{64} \, \sqrt {2} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} + 2 \, \sqrt {x}\right )}\right ) + \frac {21}{64} \, \sqrt {2} \arctan \left (-\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} - 2 \, \sqrt {x}\right )}\right ) + \frac {21}{128} \, \sqrt {2} \log \left (\sqrt {2} \sqrt {x} + x + 1\right ) - \frac {21}{128} \, \sqrt {2} \log \left (-\sqrt {2} \sqrt {x} + x + 1\right ) + \frac {7 \, x^{\frac {5}{2}} + 11 \, \sqrt {x}}{16 \, {\left (x^{4} + 2 \, x^{2} + 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(x^2+1)^3/x^(1/2),x, algorithm="maxima")

[Out]

21/64*sqrt(2)*arctan(1/2*sqrt(2)*(sqrt(2) + 2*sqrt(x))) + 21/64*sqrt(2)*arctan(-1/2*sqrt(2)*(sqrt(2) - 2*sqrt(
x))) + 21/128*sqrt(2)*log(sqrt(2)*sqrt(x) + x + 1) - 21/128*sqrt(2)*log(-sqrt(2)*sqrt(x) + x + 1) + 1/16*(7*x^
(5/2) + 11*sqrt(x))/(x^4 + 2*x^2 + 1)

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mupad [B]  time = 4.75, size = 61, normalized size = 0.47 \[ \frac {\frac {11\,\sqrt {x}}{16}+\frac {7\,x^{5/2}}{16}}{x^4+2\,x^2+1}+\sqrt {2}\,\mathrm {atan}\left (\sqrt {2}\,\sqrt {x}\,\left (\frac {1}{2}-\frac {1}{2}{}\mathrm {i}\right )\right )\,\left (\frac {21}{64}+\frac {21}{64}{}\mathrm {i}\right )+\sqrt {2}\,\mathrm {atan}\left (\sqrt {2}\,\sqrt {x}\,\left (\frac {1}{2}+\frac {1}{2}{}\mathrm {i}\right )\right )\,\left (\frac {21}{64}-\frac {21}{64}{}\mathrm {i}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x^(1/2)*(x^2 + 1)^3),x)

[Out]

2^(1/2)*atan(2^(1/2)*x^(1/2)*(1/2 - 1i/2))*(21/64 + 21i/64) + 2^(1/2)*atan(2^(1/2)*x^(1/2)*(1/2 + 1i/2))*(21/6
4 - 21i/64) + ((11*x^(1/2))/16 + (7*x^(5/2))/16)/(2*x^2 + x^4 + 1)

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sympy [B]  time = 8.46, size = 481, normalized size = 3.73 \[ \frac {56 x^{\frac {5}{2}}}{128 x^{4} + 256 x^{2} + 128} + \frac {88 \sqrt {x}}{128 x^{4} + 256 x^{2} + 128} - \frac {21 \sqrt {2} x^{4} \log {\left (- 4 \sqrt {2} \sqrt {x} + 4 x + 4 \right )}}{128 x^{4} + 256 x^{2} + 128} + \frac {21 \sqrt {2} x^{4} \log {\left (4 \sqrt {2} \sqrt {x} + 4 x + 4 \right )}}{128 x^{4} + 256 x^{2} + 128} + \frac {42 \sqrt {2} x^{4} \operatorname {atan}{\left (\sqrt {2} \sqrt {x} - 1 \right )}}{128 x^{4} + 256 x^{2} + 128} + \frac {42 \sqrt {2} x^{4} \operatorname {atan}{\left (\sqrt {2} \sqrt {x} + 1 \right )}}{128 x^{4} + 256 x^{2} + 128} - \frac {42 \sqrt {2} x^{2} \log {\left (- 4 \sqrt {2} \sqrt {x} + 4 x + 4 \right )}}{128 x^{4} + 256 x^{2} + 128} + \frac {42 \sqrt {2} x^{2} \log {\left (4 \sqrt {2} \sqrt {x} + 4 x + 4 \right )}}{128 x^{4} + 256 x^{2} + 128} + \frac {84 \sqrt {2} x^{2} \operatorname {atan}{\left (\sqrt {2} \sqrt {x} - 1 \right )}}{128 x^{4} + 256 x^{2} + 128} + \frac {84 \sqrt {2} x^{2} \operatorname {atan}{\left (\sqrt {2} \sqrt {x} + 1 \right )}}{128 x^{4} + 256 x^{2} + 128} - \frac {21 \sqrt {2} \log {\left (- 4 \sqrt {2} \sqrt {x} + 4 x + 4 \right )}}{128 x^{4} + 256 x^{2} + 128} + \frac {21 \sqrt {2} \log {\left (4 \sqrt {2} \sqrt {x} + 4 x + 4 \right )}}{128 x^{4} + 256 x^{2} + 128} + \frac {42 \sqrt {2} \operatorname {atan}{\left (\sqrt {2} \sqrt {x} - 1 \right )}}{128 x^{4} + 256 x^{2} + 128} + \frac {42 \sqrt {2} \operatorname {atan}{\left (\sqrt {2} \sqrt {x} + 1 \right )}}{128 x^{4} + 256 x^{2} + 128} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(x**2+1)**3/x**(1/2),x)

[Out]

56*x**(5/2)/(128*x**4 + 256*x**2 + 128) + 88*sqrt(x)/(128*x**4 + 256*x**2 + 128) - 21*sqrt(2)*x**4*log(-4*sqrt
(2)*sqrt(x) + 4*x + 4)/(128*x**4 + 256*x**2 + 128) + 21*sqrt(2)*x**4*log(4*sqrt(2)*sqrt(x) + 4*x + 4)/(128*x**
4 + 256*x**2 + 128) + 42*sqrt(2)*x**4*atan(sqrt(2)*sqrt(x) - 1)/(128*x**4 + 256*x**2 + 128) + 42*sqrt(2)*x**4*
atan(sqrt(2)*sqrt(x) + 1)/(128*x**4 + 256*x**2 + 128) - 42*sqrt(2)*x**2*log(-4*sqrt(2)*sqrt(x) + 4*x + 4)/(128
*x**4 + 256*x**2 + 128) + 42*sqrt(2)*x**2*log(4*sqrt(2)*sqrt(x) + 4*x + 4)/(128*x**4 + 256*x**2 + 128) + 84*sq
rt(2)*x**2*atan(sqrt(2)*sqrt(x) - 1)/(128*x**4 + 256*x**2 + 128) + 84*sqrt(2)*x**2*atan(sqrt(2)*sqrt(x) + 1)/(
128*x**4 + 256*x**2 + 128) - 21*sqrt(2)*log(-4*sqrt(2)*sqrt(x) + 4*x + 4)/(128*x**4 + 256*x**2 + 128) + 21*sqr
t(2)*log(4*sqrt(2)*sqrt(x) + 4*x + 4)/(128*x**4 + 256*x**2 + 128) + 42*sqrt(2)*atan(sqrt(2)*sqrt(x) - 1)/(128*
x**4 + 256*x**2 + 128) + 42*sqrt(2)*atan(sqrt(2)*sqrt(x) + 1)/(128*x**4 + 256*x**2 + 128)

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